# Imaginary Number in Logic

The equation $x^2 = -1$ was once said to have no solution. Then the number $i$ was discovered (or invented?) and our number system got richer. In particular, in this new wonderful world of complex numbers, we can prove the fundamental theorem of algebra, a consequence of which is that every polynomial is solvable in the complex domain.

In a similar vein, the logical statement $P = \lnot P$ has no solution in the set $\{True, False\}$. This is the well known liar's paradox, and has appeared in various forms throughout logic e.g. Russel's paradox, Godel's incompleteness theorem.

Now say we invent a new logical value $iTrue$, an imaginary one if you like, that is defined as the solution of the equation $P = \lnot P$. Our propositions would then range over $\{True, False, iTrue\}$.

My Question: Could this new system, or one like it, free us from such paradoxical, unsolvable logical statements in an analogous way to the way complex numbers freed us from unsolvable polynomials?

My Thoughts: I am skeptical of the above system. I feel that it may help us "solve" some paradoxes but not all. Furthermore it is not obvious what these solutions would mean. On the other hand, it would be lovely if the analogy with complex numbers actually worked on a more rigorous footing.

• @ColmBhandal Yes, but my point is that saying $x^2 = -1$ has a solution is completely false. The complex numbers don't give us the ability to construct a square whose area is $-1$. Commented Sep 4, 2015 at 15:11
• Right, so this is a question about how changing the underlying algebraic structure changes the solution set to an algebraic equation. Commented Sep 4, 2015 at 17:46
• what you intend to make is called fuzzy logic [high,average,low]. Commented Sep 5, 2015 at 11:41
• @ColmBhandal I'm not suggesting that all mathematical inquiries have an immediate practical application to be worthwhile. It's just that if your primary motivation is to resolve or work around various known paradoxes, there are easier, more intuitve ways to do it. Commented Sep 5, 2015 at 16:04
• $iTrue$ is probably owned by Apple, so try something else. Commented Sep 6, 2015 at 17:10

There is a vast literature on many-valued logics. Lukasiewicz's original 3-valued logic is perhaps the simplest such logic and the extra truth value $P$ satisfies your formula $P \iff \lnot P$. Lukasiewicz developed this into what are now called Lukasiewicz logics that have been intensively studied and generalised over the years. One of these generalisations is the subject of fuzzy logic, which has practical applications, e.g. to model situations where knowledge is imperfect.

• "Possibly"- that's exactly the kind of solution I was looking for! Cheers. Commented Sep 5, 2015 at 14:58

You may be underestimating what was involved in the creation of the complex number system. It's not just a matter of introducing a symbol $$i$$ and declaring that $$i^2=-1$$. Rather, it's important that much of what people knew about real numbers also applies to complex numbers; specifically, $$\mathbb C$$ is a field of characteristic zero (though not an ordered field). So people could continue to manipulate equations (though not inequalities) involving complex numbers just as they did with real numbers. It turned out that the properties of real numbers that persist when one passes to complex numbers are enough to give an interesting and useful theory (and later, people found additional properties of $$\mathbb C$$, like algebraic closure, that make it even more interesting and useful).

Analogously, it would do little good to introduce a new truth value and declare it to be equal to its negation. One would need to show that the new system of truth values retains enough of the properties of the traditional system so that people can continue to reason more or less as they are accustomed to do.

• Q: Is it good to introduce a truth value equal to its negation? A: Maybe...? Commented Sep 22, 2018 at 17:32
• @AsafKaragila your comment has two levels, which is nice. Yes if we defined 1 as true and 0 as false, with maybe as 0.5, and negation as $1 - x$, then we can make something that works. Commented Sep 24, 2018 at 11:02
• Andreas yeah to claim that the complex number system was simply a gimmick to solve that one tiny equation is probably a bit of an overstatement. It's probably more correct to say it was the only way to develop a fundamental theorem of algebra. Commented Sep 24, 2018 at 11:04

As Rob Arthan says, there's a vast literature on this topic, only some of it in application to the paradoxes.

As a solution to paradoxes like the liar, though, the introduction of a third truth-value meets an immediate difficulty, the so-called "strengthened liar":

This sentence is not true.

• Very good. So if we represented truth/falsehood on a fuzzy scale e.g. the interval $[0, 1]$, with true as $1$, then any value in the interval $[0, 1)$ would imply the value is $1$, which would in turn imply the value is in $[0, 1)$, and so we're back to paradox land. Hence my skepticism in the question. Commented Sep 6, 2015 at 14:00
• In practice, mathematicians don't define function symbols everywhere, and probably mostly not predicate symbols either. E.g., the slope of vertical lines is "undefined"; using classical logic, one must choose between saying (dishonestly) that lines are perpendicular if and only if their slopes are negative reciprocals or (somewhat pedantically) that oblique lines are perpendicular if and only if their slopes are negative reciprocals. Using my favorite 3-valued logic one can honestly and non-pedantically say that "lines be perpendicular if true and only if their slopes be negative reciprocals". Commented Sep 6, 2015 at 16:49

Your introduction to the question seems to be very similar than the thoughts of L. H. Kauffman in Virtual Logic and Imaginary Values. Either you were inspired by them or it could be worth checking them.

Regarding your question, it seems that you are looking for a logic where the Law of Excluded Middle is not an axiom. So, I would suggest to check the field of paraconsitent logic.